Lesson Plan: Identifying Proportional Relationships

Lesson Summary

This lesson focuses on helping students recognize and represent proportional relationships between quantities. They will learn to identify proportional relationships in tables and graphs, determine the constant of proportionality, and understand the characteristics of linear relationships that pass through the origin.

Lesson Objectives

This lesson can be completed in one 50-minute class period but may require additional time depending on your class.

• Recognize proportional relationships in tables and graphs
• Determine if a relationship is proportional
• Identify the constant of proportionality

Common Core Standards

• 7.RP.A.2 Recognize and represent proportional relationships between quantities.
• 7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
• 7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Prerequisite Skills

• Understanding of ratios and unit rates
• Basic graphing skills

Key Vocabulary

• Proportion: An equation stating that two ratios are equal.
• Constant of Proportionality: The constant ratio between two proportional quantities, often represented by 'k' in the equation y = kx.
• Origin: The point (0,0) on a coordinate plane where the x-axis and y-axis intersect.
• Linear Relationship: A relationship between two variables that, when graphed, forms a straight line.

Multimedia Resources

• A collection of definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/Definitions--RatiosProportionsPercents
• A student tutorial slide show on definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/library/slideshow/student-tutorial-ratios-proportions-and-percents-definitions

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Equivalent Ratios

Identify equivalent ratios in a table

Example: Given the following table of values, identify pairs of numbers that form equivalent ratios:

Solution: All pairs form equivalent ratios (1:3), as y is always 3 times x. You can use this Desmos activity to graph the coordinates and find the line of best fit:

https://www.desmos.com/calculator/gazl6xwvfg

Activity 2: Review of Linear Functions:

Begin by revisiting the concept of linear functions. Discuss how a linear function represents a constant rate of change between two variables and can be expressed in the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Use this slide show to review linear functions:

https://www.media4math.com/library/slideshow/domain-and-range-functions-linear

Activity 3: Review of Graphs of Linear Functions:

Examine the graphical representation of linear functions. Highlight how the graph of a linear function is a straight line, and discuss the significance of the slope and y-intercept in determining the line's direction and position. Emphasize that for proportional relationships, the graph will be a straight line passing through the origin (0,0).

Teach

Introduction to Proportional Relationships

A proportional relationship exists when two quantities increase or decrease at the same rate. This means that their ratio remains constant, and they can be represented by an equation of the form:

$$y = kx$$

where:

• $y$ is the dependent variable.
• $x$ is the independent variable.
• $k$ is the constant of proportionality.

Key characteristics of proportional relationships include:

• The ratio $\frac{y}{x}$ remains constant.
• The graph of the relationship forms a straight line that passes through the origin (0,0).
• The constant of proportionality $k$ determines the steepness (slope) of the line.

Recognizing Proportional Relationships in Tables

A table represents a proportional relationship when the ratio between corresponding values of $x$ and $y$ is always the same. For example:

Since the ratio remains constant ($k = 3$), this table represents a proportional relationship.

Graphing Proportional Relationships

The graph of a proportional relationship is always a straight line that passes through the origin. The slope of this line is equal to the constant of proportionality $k$. For example, the equation:

$$y = 2x$$

Produces a line that passes through (0,0), (1,2), (2,4), and (3,6), all maintaining the same ratio.

Interpreting the Constant of Proportionality

The constant of proportionality $k$ can be interpreted based on the context. For example:

• If $k = 5$ in $y = 5x$, it may represent "5 dollars per hour" or "5 miles per gallon," depending on the situation.
• In a recipe, if $k = 2$, it means "for every 1 cup of sugar, 2 cups of flour are needed."

Understanding these key ideas allows students to apply proportional reasoning in different contexts and problem-solving situations.

Definitions

• Proportion: An equation stating that two ratios are equal
• Constant of proportionality: The constant ratio between two proportional quantities
• Origin: The point (0,0) on a coordinate plane
• Linear relationship: A relationship that forms a straight line when graphed

Use this slide show to review these other related definitions:

https://www.media4math.com/library/slideshow/definitions-proportions-and-proportional-relationships

Example 1. Science: Hooke's Law (Force and Spring Extension)

This slide show discusses Hooke's Law in detail. 

https://www.media4math.com/library/slideshow/application-linear-functions-hookes-law

 Use it as background to frame the following problem solving scenario:

A physics student is investigating the relationship between the force applied to a spring and its extension. She records the following measurements:

• Graph: Plot points and observe direct proportion. Use this Desmos activity:
  https://www.desmos.com/calculator/lr5b4efzgq
•  Constant of proportionality: Extension/Force = 0.5 cm/N
• Explain how this demonstrates a direct proportional relationship.
• Discuss real-world applications, such as in the design of suspension systems or measuring instruments.

Example 2. Engineering: Gear Ratios

Use this slide show to demonstrate this problem solving scenario with gear ratios:

https://www.media4math.com/library/slideshow/applications-gear-ratios

 Here is a summary of the scenario

An engineer is designing a gear system for a new machine. She needs to determine the relationship among the number of teeth for each of the gears:

• Gear A has 20 teeth
• Gear B has 12 teeth
• Gear C has 8 teeth

Determine the number of turns gear C has to make in order for gears A and B complete at least one turn. 

• The gear ratio, in simplified form is this: 5:3:2
• The revolution ratio, in simplified form is this: 2:3:5
• Gear C must complete at least 2.5 turns for gears A and B to complete at least one turn

Example 3. Art: Color Mixing

Word problem: An artist is creating a new shade of green by mixing yellow and blue paint. He wants to ensure he can consistently reproduce this color:

• Graph: Plot points and observe direct proportion
• Constant of proportionality: Blue/Yellow = 0.4
• Explain how this is used in creating consistent shades of green
• Discuss applications in graphic design, painting, and digital art

Review

Lesson Summary

In this lesson, students explored proportional relationships and how to identify them using tables, graphs, and equations. They learned to:

• Determine whether a relationship is proportional by checking if the ratio $\frac{y}{x}$ is constant.
• Recognize that proportional relationships are represented by equations in the form $y = kx$, where $k$ is the constant of proportionality.
• Interpret proportional relationships using tables and graphs, ensuring that the graph is a straight line passing through the origin (0,0).
• Apply proportional reasoning to real-world contexts, such as unit rates, scaling, and financial calculations.

By mastering these concepts, students gained a deeper understanding of how proportional relationships work and how to apply them to problem-solving scenarios.

Key Vocabulary

• Proportion: An equation stating that two ratios are equal.
• Constant of Proportionality: The constant ratio between two proportional quantities, represented by $k$ in the equation $y = kx$.
• Origin: The point (0,0) on a coordinate plane where the x-axis and y-axis intersect.
• Linear Relationship: A relationship between two variables that, when graphed, forms a straight line.
• Unit Rate: A rate where the denominator is 1 (e.g., miles per hour, cost per item).

Additional Examples

Example 1 (Business): Sales Commission

A real estate agent earns a 5% commission on each house sale. The following table shows the commission earned for different house prices:

Ask students to:

1. Determine if this is a proportional relationship
2. Identify the constant of proportionality
3. Calculate the commission for a \$350,000 house sale

Solutions:

1. Yes, this is a proportional relationship. The ratio of commission to house price is constant (1:20 or 0.05).
2. The constant of proportionality is 0.05 or 5%.
3. Commission for \$350,000 sale: 350,000 * 0.05 = \$17,500

Example 2 (Sports): Running Pace

A runner is training for a marathon and records her distance and time for several runs:

Ask students to:

1. Determine if this is a proportional relationship
2. Identify the constant of proportionality (pace in minutes per mile)
3. Predict the time for a 13-mile run (half marathon)

Solutions:

1. Yes, this is a proportional relationship. The ratio of time to distance is constant (8:1).
2. The constant of proportionality is 8 minutes per mile.
3. Time for a 13-mile run: 13 * 8 = 104 minutes or 1 hour and 44 minutes

Quiz

Answer the following questions.

1. Is the relationship between x and y proportional?

   x	y
   2	6
   4	12
   6	18
   8	24

2. What is the constant of proportionality in the relationship from question 1?
   
    
3. Does the graph of y = 2x + 1 represent a proportional relationship?
   
    
4. If a car travels 240 miles in 4 hours at a constant speed, what is the constant of proportionality?
   
    
5. In the equation y = kx, what does k represent?
   
    
6. Is the origin always included in the graph of a proportional relationship?
   
    
7. If 3 shirts cost $24, how much would 5 shirts cost in this proportional relationship?
   
    
8. What is the constant of proportionality if 8 ounces of a liquid occupy 10 cubic inches?
   
    

9. Does the table represent a proportional relationship?

   x	y
   0	3
   2	5
   4	7
   6	9

    

10. If y is proportional to x and y = 15 when x = 3, what is the constant of proportionality?

Answer Key

1. Yes
2. 3
3. No
4. 60 miles per hour
5. The constant of proportionality
6. Yes
7. $40
8. 1.25 cubic inches per ounce
9. No
10. 5

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